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The Hadamard transform H m is a 2 m × 2 m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2 m real numbers x n into 2 m real numbers X k.The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices n and k.
Let H be a Hadamard matrix of order n.The transpose of H is closely related to its inverse.In fact: = where I n is the n × n identity matrix and H T is the transpose of H.To see that this is true, notice that the rows of H are all orthogonal vectors over the field of real numbers and each have length .
In computational mathematics, the Hadamard ordered fast Walsh–Hadamard transform (FWHT h) is an efficient algorithm to compute the Walsh–Hadamard transform (WHT). A naive implementation of the WHT of order n = 2 m {\displaystyle n=2^{m}} would have a computational complexity of O( n 2 {\displaystyle n^{2}} ) .
Simon's problem considers access to a function : {,} {,}, as implemented by a black box or an oracle. This function is promised to be either a one-to-one function, or a two-to-one function; if is two-to-one, it is furthermore promised that two inputs and ′ evaluate to the same value if and only if and ′ differ in a fixed set of bits. I.e.,
Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity. For N = 2 , 3 {\displaystyle N=2,3} and 5 {\displaystyle 5} all complex Hadamard matrices are equivalent to the Fourier matrix F N {\displaystyle F_{N}} .
Example: The Hadamard transform on a 3-qubit register | . Here the amplitude for each measurable state is 1 ⁄ 2 . The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases.
The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression.It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison.
The Kronecker product of two Hadamard matrices of sizes m and n is an Hadamard matrix of size mn. By forming Kronecker products of matrices from the Paley construction and the 2 × 2 matrix, = [], Hadamard matrices of every permissible size up to 100 except for 92 are produced.